Hún tilkynnir vinnufélaganum sínum um nýja frestinn.

Because tilkynna commonly takes a recipient in the dative: tilkynna einhverjum = inform someone / notify someone.
So vinnufélaganum is dative singular definite of vinnufélagi (coworker), meaning to the coworker.

It’s the combination of:

• the definite article suffix (the -inn/-inn/-ið system, which changes by case), and
• the dative singular ending.

In other words, vinnufélaganum = to the coworker (not just to a coworker).

sínum is the reflexive possessive (from sinn) and it refers back to the subject (Hún).
So vinnufélaganum sínum means to her (own) coworker.

If you used hennar, it would typically suggest the coworker belongs to some other female person already being talked about (i.e., not reflexive to the subject), depending on context.

Because it must agree with the noun it modifies (vinnufélaganum):

• gender: masculine (vinnufélagi)
• number: singular
• case: dative (because of tilkynna einhverjum)

So the matching form is sínum (masc. dat. sg.).

um introduces what the notification is about: tilkynna einhverjum um eitthvað = inform someone about something.
The preposition um governs the accusative, so frestinn is in the accusative.

Because um takes the accusative here.
The base dictionary form is frestur (masculine). Its accusative singular definite form is frestinn.

A quick mini-paradigm (singular, with definite article) is roughly:

• nominative: fresturinn
• accusative: frestinn
• dative: frestinum
• genitive: frestsins

Because frestinn is definite (the deadline), Icelandic usually uses the weak adjective form with definite nouns.

So you get:

• nýja frestinn = the new deadline (weak adjective)

Whereas with an indefinite noun you’d typically use the strong form:

• nýjan frest = a new deadline (strong adjective)

tilkynnir is:

• present tense
• 3rd person singular
• from the verb tilkynna (to notify, announce, inform)

So it matches Hún (she): Hún tilkynnir = she notifies/informs.

This is a very neutral, common order: Subject – Verb – Indirect object – Prepositional phrase
Hún (S) tilkynnir (V) vinnufélaganum sínum (IO) um nýja frestinn (PP)

You can move elements for emphasis, but Icelandic still follows the V2 rule in main clauses (the finite verb tends to be in the second position). For example, you could front the topic:

• Um nýja frestinn tilkynnir hún vinnufélaganum sínum.
  This emphasizes about the new deadline.

A rough guide:

• Hún ≈ hoon (with a long ú)
• tilkynnir ≈ til-kin-ir, but with Icelandic ky sounding like a palatal k (closer to kyi than ku/kee)
• vinnufélaganum ≈ VINN-uh-fyelay-uh-num
• sínum ≈ SEE-num
• um ≈ um (short)
• nýja ≈ NEE-ya (ý is like a long “ee” sound)
• frestinn ≈ FRES-tin (short i)

(Exact pronunciation varies by speaker, but stress is generally on the first syllable of Icelandic words.)